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Question: The decomposition reaction of \[S{{O}_{2}}C{{l}_{2}}\] is the first order reaction. It’s concentrati...

The decomposition reaction of SO2Cl2S{{O}_{2}}C{{l}_{2}} is the first order reaction. It’s concentration halves in 60 minutes, what is its rate constant value?
(A) 0.1155 min⁡ute−1\min ut{{e}^{-1}}
(B) 0.07676 min⁡ute−1\min ut{{e}^{-1}}
(C) 0.7676 min⁡ute−1\min ut{{e}^{-1}}
(D) 0.01155 min⁡ute−1\min ut{{e}^{-1}}

Explanation

Solution

Hint: Here decomposition means breaking ofSO2Cl2S{{O}_{2}}C{{l}_{2}} is a first order means its rate of decomposition depends on single power of concentration.

Step by step solution:
Decomposition reaction of SO2Cl2S{{O}_{2}}C{{l}_{2}}
SO2Cl2(g)→SO2(g) + Cl2(g)S{{O}_{2}}C{{l}_{2}}(g)\to S{{O}_{2}}\left( g \right)\text{ }+\text{ }C{{l}_{2}}\left( g \right)
We know that for a 1st order reaction,
r=k[C]r=k[C]
Where r is rate of reaction,
‘k’ is the rate constant
‘C’ is the concentration of SO2Cl2S{{O}_{2}}C{{l}_{2}}
And we know that for 1st order reaction:
t1!/!2  = 0.693k {{t}_{{\scriptscriptstyle 1\\!/\\!{ }_2}}}~\text{ }=\text{ }\dfrac{0.693}{k}~
Where t1!/!2  {{t}_{{\scriptscriptstyle 1\\!/\\!{ }_2}}}~\text{ }is half life of reaction.
It is given that t1!/!2  {{t}_{{\scriptscriptstyle 1\\!/\\!{ }_2}}}~\text{ }= 60 min
Now we put this given value of half life in above formula and rearranging:
k = 0.69360 \text{k }=\text{ }\dfrac{0.693}{60}~
= 0.693 / 6
k= 0.01155 min⁡−1 {{\min }^{-1}}~
k= 1.1925 ×10−4sec⁡−1\times {{10}^{-4}}{{\sec }^{-1}}
Alternative method
We can also solve this problem by using the first order reaction equation, which is
k=2.303tlog⁡[C0][C]k=\dfrac{2.303}{t}\log \dfrac{[{{C}_{0}}]}{[C]}
And we know that at t1!/!2  {{t}_{{\scriptscriptstyle 1\\!/\\!{ }_2}}}~\text{ }value of [C] = [C0][{{C}_{0}}]/2
Now we will put value of t1!/!2  {{t}_{{\scriptscriptstyle 1\\!/\\!{ }_2}}}~\text{ }value of [C] in above equation:
k=2.30360log⁡[C0]([C0]2)k=\dfrac{2.303}{60}\log \dfrac{[{{C}_{0}}]}{(\dfrac{[{{C}_{0}}]}{2})}
k= 0.01155 min⁡−1 {{\min }^{-1}}~
k= 1.1925 ×10−4sec⁡−1\times {{10}^{-4}}{{\sec }^{-1}}
So, from the above calculation we can see that the correct answer is option “D”.

Note: Options can be both in units of minutes or seconds. But in this question options are given in a minute unit so no need to convert the value of ‘k’ in seconds. From units of ‘k’ we can also check the order of reaction.